What is the angle at which a ball falls off a hemisphere?

AI Thread Summary
The problem involves determining the angle at which a ball slides off a frictionless hemisphere after starting from rest at the top. The forces acting on the ball can be resolved into components, with gravitational force split into mg sin(theta) and mg cos(theta). The key to solving the problem is finding the point where the centripetal force equals zero, indicating the normal force is no longer acting on the ball. Both calculus and energy methods can be used to derive the angle theta. Ultimately, the solution requires analyzing the forces and energy conservation to find the critical angle for the ball's departure from the hemisphere.
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Homework Statement


A ball with initial velocity 0 and mass m starts at the top of a hemisphere and begins to slid down the frictionless side. The radius of the hemisphere is r. At what angle does the ball fall off the hemisphere?

Homework Equations


mgsintheta, mgcostheta, fn
mv^2/r

The Attempt at a Solution


I guess that you can split mg into mgsin theta and mg cos theta and that one of them would create centripetal force. Then you can find where the centripetal force is 0 because there would be no normal force when the ball falls off? I'm not sure. Also, my teacher mentioned that there is a way to solve this with and without calculus.
 
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Find the place (or time) at which the normal force becomes zero, if you constrain the ball is constrained to remain on the surface of the hemisphere.
 
you can use energy to find theta, solve v then substitute into your force equation
 

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